{
  "abstract": "Arrow patterns, introduced by Berman and Tenner, provide a unified framework for studying permutation classes where both one-line and cycle structure constraints are present. In this paper, we continue the systematic study of arrow pattern avoidance initiated by Archer and Laudone. We establish several structural results, including a key lemma that translates arrow patterns into vincular patterns under certain conditions, and derive a series of arrow-Wilf equivalences arising from reversal, complementation, and insertion operations. We also resolve the two cases $(12;3\\to 3)$ and $(21;3\\to 3)$ left open by Archer and Laudone, and enumerate the arrow patterns of the form $(ν; b\\to c)$ of size $3$ with $ν\\in \\{31, 23, 32\\}$ and $b,c\\in [3]$, providing explicit formulas connecting the results to Bell numbers, Bessel numbers, Catalan numbers, and derangement numbers. Together with earlier work of Archer and Laudone, this leaves only $(32;1\\to 3)$ unresolved for $|ν|\\le 2$, which we pose as an open problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29392",
  "author_total": 2,
  "authors": [
    {
      "name": "Robin D. P. Zhou"
    },
    {
      "name": "Xinyang Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xsm4nt9ypt62.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29392",
  "fulltext_url": null,
  "id": "0xsm4nt9ypt62",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29392",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7214377203",
  "pdf_url": "https://arxiv.org/pdf/2609.29392",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xsm4nt9ypt62.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.29392",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29392"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Arrow-Wilf equivalences and enumerative results for short arrow patterns",
  "title_text": "Arrow-Wilf equivalences and enumerative results for short arrow patterns",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/0xsm4nt9ypt62",
  "venue": "arXiv",
  "versions": []
}